This paper discusses the algebraic structure of a semiring, with additional attention given to the special case of a dioid. The set $R$ is a semiring for the binary internal laws $\oplus$ and $\otimes$ if $(R,\oplus)$ is a commutative monoid with the neutral element $0$, $(R,\otimes)$ is a monoid with the neutral element $1$, $\otimes$ is left- or right-distributive with respect to $\oplus$ and if $0$ is absorbing for $\otimes$, i. e. $\forall a \in R; a \otimes 0 = 0\otimes a = 0$. If additionally $(R,\oplus)$ is also ordered with the canonical order relation $\le$, we instead call $(R,\oplus,\otimes)$ a dioid. Both terms have existed for some time now and most of the classical questions relating to the structures, from the perspective of linear algebra, have already been answered. From among those, this paper will present some of the more elementary results. In particular, the focus will be on properties of semirings and dioids and on generalizations of concepts from classical linear algebra over fields such as the existence and properties of bases of an $R$-semimodule, the properties and invertibility of a matrix over a semiring $R$ and the Cayley-Hamilton theorem.
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